Curiosities regarding waiting times in P\'{o}lya's urn model

TL;DR

This paper investigates the waiting times in a Pólya's urn model, revealing surprising properties about the finiteness of moments depending on initial conditions, with implications for understanding stochastic processes.

Contribution

It provides a detailed analysis of the moments of waiting times in Pólya's urn, establishing conditions for finiteness of expectation and variance based on initial urn composition.

Findings

Expected waiting time is infinite for the initial scenario (b=w=1).

Expected waiting time is finite if b≥2, regardless of w.

Variance of waiting time is finite if b≥3, regardless of w.

Abstract

Consider an urn initially containing $b$ black and $w$ white balls. Select a ball at random and observe its color. If it is black, stop. Otherwise, return the white ball together with another white ball to the urn. Continue selecting at random, each time adding a white ball, until a black ball is selected. Let $T_{b,w}$ denote the number of draws until this happens. Surprisingly, the expectation of $T_{b,w}$ is infinite for the "fair" initial scenario $b =w=1$, but finite if $b=2$ and $w=10^9$. In fact, $\mathbb{E}[T_{b,w}]$ is finite if and only if $b\ge 2$, and the variance of $T_{b,w}$ is finite if and only if $b \ge 3$, regardless of the number $w$ of white balls. These observations extend to higher moments.